Impossibility of Succinct Quantum Proofs for Collision-Freeness
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Aaronson_Impossibility of Succinct Quantum Proofs for Collision-Freeness.pdf
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Author(s)
Aaronson, Scott
Date Issued
January 2011
Journal
Electronic Colloquium on Computational Complexity
Publisher
Hasso-Plattner-Institut für Softwaresystemtechnik GmbH
Citation
Aaronson, Scott. "Impossibility of Succinct Quantum Proofs for Collision-Freeness." Electronic Colloquium on Computational Complexity (2011): TR11-001.
Version
Author's final manuscript
Abstract
We show that any quantum algorithm to decide whether a function f:\left[n\right] \rightarrow\left[ n\right] is a permutation or far from a permutation\ must make \Omega\left( n^{1/3}/w\right) queries to f, even if the algorithm is given a w-qubit quantum witness in support of f being a permutation. This implies that there exists an oracle A such that \mathsfSZKA\mathsfQMAA , answering an eight-year-old open question of the author. Indeed, we show that relative to some oracle, \mathsfSZK is not in the counting class \mathsfA\mathsf0\mathsfPP defined by Vyalyi. The proof is a fairly simple extension of the quantum lower bound for the collision problem..
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
http://eccc.hpi-web.de/report/2011/001/
